and \(\lbrace \psi _{\bf k}\rbrace \)  and \(\lbrace \lambda _{{\bf k}}\rbrace \)  are a complete set of orthonormal eigenfunctions and a set of corresponding eigenvalues of the Laplace operator \(-\nabla ^{2}\)  in \(\mathbb {R}^{3}\) . Notice that each \(\psi _{\bf k}({\bf R})\)  must satisfy the Neumann boundary equation \(\left.\nabla \psi _{{\bf k}}\right|_{{\bf R}\in \partial \mathcal {D}}=0\) . In addition, it is known [1]}, [2]} that for Neumann boundary Laplacian eigenvalue problem there is a zero eigenvalue \(\lambda _{0}=0\)  corresponding to a positive eigenfunction given by \(\psi _{0}={1/\sqrt{V}}\) .
